No, top roping is not a part of any other kind of climbing other than free climbing in general.

Though a big part of the argument for sport climbing - as opposed to trad climbing with gear - is usually that it allows you to dispense with dicking around with all the gear and just focus on 'movement'. But given most, but certainly not all, sport climbs are single pitch climbs - and if 'movement' is really your focus - then why dick around with the clipping? Just TR it and dispense with the clipping nonsense as well. But, there's obviously great mystery around sport climbing and the distinction between the two involves a lot of really deep semantics. (But kidding aside, there are a lot of places that don't have cliff top access for setting up TRs).

In general, there's aid climbing and free climbing. Within free climbing there's trad climbing and sport climbing. The major distinction actually isn't that one uses gear and the other uses bolts, however. The broader, and essential, difference it tactics: in sport climbing you hang on the rope resting after you fall off and then get back on the rock (derogatorily known as 'hangdogging'), in trad climbing if you fall off you lower to the ground or to the anchor on multipitch and go again with purest pulling their rope and releading and less-than-purists just going again without pulling the rope.

These days a lot of folks crossing from sport to trad bring those sport tactics of hanging over to trad climbing essentially sport climbing on gear - kind of pointless (and somewhat dangerous) and better referred to a 'sprad' climbing.

Within free climbing there's trad climbing and sport climbing. The major distinction actually isn't that one uses gear and the other uses bolts, however. The broader, and essential, difference it tactics: in sport climbing you hang on the rope resting after you fall off and then get back on the rock (derogatorily known as 'hangdogging'), in trad climbing if you fall off you lower to the ground or to the anchor on multipitch and go again with purest pulling their rope and releading and less-than-purists just going again without pulling the rope.

And, believe it or not, there are people who actually think that distinction is important.

For now, let's assume we're talking about rock climbing. (There's also alpine climbing, ice climbing, and mixed climbing. Alpine is, generally, associated with making it up peaks, and may include technical rock and ice climbing, but often is extended scrambling and usually dealing with various snow conditions, etc. Ice climbing is climbing near vertical to overhanging ice with ice axes and pitons. Mixed is climbing a mixture of ice and rock with ice axes and pitons.)

First major break-down: aid vs free climbing.

In aid, you use gear to make progress up the rock -- whether it is clipping a bolt and pulling on it, hauling up on a rope from above, or placing nuts or cams, and stepping up them on etriers.

In free climbing all progress up the rock is achieved using only your own body -- primarily hands and feet, but knees, elbows, butts, shoulders, or whatever are all fine. Free climbing is generally sub-divided by how you protect yourself in case of a fall. They are:

Free-soloing: no rope, no gear, no protection. If you fall, severe injury and death are to be expected.

Bouldering: no rope, but restricted height to limit risk from fall. Often pads (cushioning) in the landing zone to reduce impact force and protect. Often a spotter or spotters to help control the fall. (High-ball bouldering is bouldering to a greater height than might be normal, increases the risk -- the line between high-ball bouldering and Free-soloing is very fine.)

Deep-water soloing: Essentially bouldering over water deep enough that a fall will be cushioned/protected by landing in water, rather than on something solid.

Leading on gear (often called Trad climbing): rope is attached to the climber, and as the climber progresses up the wall, she attaches gear -- mechanical devices range from simple (nuts, hexes) to complex (spring-loaded camming devices) -- and then attaches the rope to the gear, with hope that on a fall, the gear will stay attached to the wall, and limit the distance of the fall. (Fall generally will be amount of distance above the most recent protection point x2, plus slack in the system, plus rope stretch.)

Leading on bolts (often called Sport climbing): rope is attached to the climber, and he progresses up the wall, he clips the rope to bolts that are emplaced in the rock. Generally easier to do, and taking far less time than place gear.

Top-rope climbing: the rope is anchored at the top of the climb, and the rope is always above the climber with minimal slack. Falls are (generally) of the most limited length, and smallest impact force, in this case. Fall will be generally slack in the system (very little to none) plus rope stretch.

TopRope is a binary adjective to describe a climb. It is not exclusive from Trad or Sport. You could TopRope on bolts or pro, though the latter would be a poor idea. Most often, a TR will be on bolts, underneath florescent lights, in a gym.

TopRope is essentially the opposite of Lead. Wikipedia has a good index of climbing terms that covers all this.

Unless your guidebook specifically mentions you can walk-up to the top to set up a TopRope, you should assume someone must send the route first on lead, and establish an anchor to TopRope from. You're never just going to find a TR dangling there, waiting all alone for you outside.

The first time I climbed outside, I walked up and tied webbing to a couple sturdy-looking trees to form my anchor. I set up a TopRope, belayed myself down, and took turns climbing with my partner.

TopRope usually elicits an upturned nose from elitists, because it's more inviting to n00bs. TopRope makes for good exercise though, and it is almost always how professional competitions are held because no time or energy is wasted clipping.

TopRope is particularly worthwhile if your gym has good autobelays that you don't have to wait on/coordinate with.

Just go out and climb ... Its sunny and just above frezing today yipeee!!!

To the OP ... It only matters for people who wont lead ... If you never lead youll be very limited in what you can do and will usually be dependant on others to lead up for you in the many climbing areas that dont have top rope access for most climbs

If you want to climb the good stuff you need to lead or be a biatch to someone who does and hope they dont get tired of it/you

I would as well and would go so far as to say that on steep, overhanging rock where you can't get back on, sport leading is actually easier. There's no hanging out resting and sorting out the moves from a TR on steep rock. On that kind of terrain TR has more in common with DWS as you're either climbing or flying. That means you have to sort out the moves while actually climbing as opposed to resting while you eyeball it all and give various combinations a go.

1 - When a climber has a toprope, that signifies that the rope and the anchor are above them, rather than below. This can happen in any type of roped climbing in which there is an anchor. (Sport, trad, ice, aid, some mountaineering, etc).

2 - If a climb is set up such that every climber who does it is on toprope, that is a distinct style called "toproping". That style is distinct from any other roped climbing with anchors, in that there is never a leader or a follower.

Top roping just means that the rope is running through a fixed anchor. In some cases the rope can be set by hiking to the top of the cliff, but in most cases it means the climb must be lead first in order to get the rope up to the fixed anchors. I've TR'd ice climbs, crack climbs, and sport climbs.